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We address the end-effector full-pose tracking control problem in free-floating space manipulators, experiencing constant non-zero linear and angular momentum. The aim is to develop an output-tracking (workspace) control law free of singularities due to parameterizing the end-effector motion and being robust against singularities of the input-output decoupling matrix (generalized Jacobian matrix). Space manipulators are modelled as open-chain multi-body systems with single- and multidegree-of-freedom joints, whose kinematics and dynamics are formulated on the Special Euclidean group SE(3). Such systems exhibit conserved (not necessarily zero) total momentum when operating in the free-floating regime, which we use to systematically reduce their dynamical equations by eliminating the base spacecraft’s motion. To avoid parameterizing the end-effector motion, we consider its full pose as the system output and develop a novel feedback linearization technique on the matrix Lie group SE(3) in the reduced phase space of the space manipulator. We then propose an intrinsic feedforward, feedback proportional-integral-derivative workspace controller involving a coordinate-free pose error function on SE(3) and velocity error on its Lie algebra. Using a Lyapunov candidate, this controller is proven to stabilize the end-effector pose to a feasible desired trajectory. The input-output decoupling matrix in the proposed control law can lose rank at some regions of the configuration space; hence, we implement a singularity-robust inverse, derived from the damped least squares method, to avoid impractical joint torques in these regions. The developed controller is implemented on a 7-degree-of-freedom manipulator onboard a spacecraft and its efficacy and robustness are demonstrated trough series of simulations.

In a debris mitigation mission, a crucial phase of the proximity operation for a space manipulator is chasing a capture point on a noncooperative target satellite. Knowing the uncertain position and velocity of the target, a learning-based online trajectory planner offers a robust solution to the chasing problem. This paper uses the concept of transfer learning to develop an online trajectory generator for the task of capturing a moving target with an uncertain space manipulator. We divide this complex task into multiple sub-tasks and order them based on their difficulty level. We employ the Deep Deterministic Policy Gradient (DDPG) algorithm to learn each sub-task individually. The DDPG is a deep reinforcement learning approach that provides the ability to work with continuous states and actions by approximating the action-value function and the policy with neural networks. We propose a novel method to transfer the knowledge gained in an easier sub-task to a more difficult one in the form of expert policy and transition memories. State and action representation has a crucial impact on learning performance, which is comprehensively studied in this paper for the task of capturing a moving target. Considering the learning performance, we show the existence of an optimal state representation, which is not necessarily the minimal representation of the system. We compare different action representations of a manipulator, i.e., joint space and workspace velocities, and demonstrate the superiority of the workspace actions. Finally, the developed transfer learning approach is implemented on a planar space manipulator with an onboard 2-link arm to generate trajectories that can capture a target randomly moving with the maximum speed of the manipulator’s end effector. To show the efficacy of the approach, its results are compared with the case where the agent learns the task from scratch.

Accurate wheel slip estimation facilitates Wheeled Mobile Robots (WMRs) with improved localization and traversability monitoring which are crucial to their autonomy in challenging environments. Although distributed track-level fusion offers better computational efficiency, often sensor-level fusion is adopted in slip estimators. Extending upon our earlier work, we develop a novel explicit track-to-track fusion algorithm for UKF-based multi-sensor networks that has immediate application to slip ratio estimation in WMRs. The algorithm demonstrates better consistency compared to the existing fusion techniques, since it implements an optimal fusion rule to sequentially combine local tracks. It also saves computational power due to the recursive propagation of cross-covariance matrices based on the statistical linearization technique, instead of performing online optimizations. We prove that this recursion only requires the information of the first and last tracks in a sequence if all local tracks are unbiased. We rigorously study various key properties of the developed fusion algorithm and show its superior level of confidence when compared to the two prominent fusion methods of sequential and batch covariance intersection. The slip ratio estimation in a six-wheel WMR is considered as a case study, where the steerable wheel sets act as a network of sensors. The proposed slip estimator works based on the rigid body kinematics and readings of purely proprioceptive sensors, i.e., an inertial measurement unit and encoders. The slip estimator's performance is evaluated in a high-fidelity software-in-the-loop simulation environment connecting MATLAB and CM Lab's Vortex Studio software. In a comparison study that considers four rival strategies, we show the superiority of the proposed fusion method offering a balance between consistency, accuracy, and speed in real-time slip estimations.

In this paper, we investigate the motion of wheeled mobile robots on rough terrains modelled as noisy nonholonomic constraints. Such constraints are the natural extension of ideal nonholonomic constraints when the Stratonovich process is directly introduced in the constraint equations. The resulting stochastic model can capture motion on rough surfaces, random skip/uncertainty in the wheel-ground point of contact, or stochastic motion of the surface. We study a differential robot with ideal noisy and affine noisy constraints, where each case models a certain aspect of motion on rough terrains. We then qualitatively investigate their corresponding stochastic dynamics through Monte-Carlo simulations. The proposed stochastic model for roving rough terrains has the potential to serve as the process model in model-based motion estimators relying on measurements from an interoceptive suite of sensors. The challenge will be dealing with the nonlinear appearance of the noise in the equations of motion.

This letter addresses the problem of robust finite-time tracking control with prescribed performance for robot manipulators experiencing uncertain inertia, external disturbance, and actuator fault. We develop a control strategy that incorporates the nonlinear H∞ concept into the backstepping approach, using a novel virtual control, to guarantee practical finite-time convergence to a trajectory, whilst the closed-loop L2 gain is less than a prespecified value. We also use adaptive gains, instead of complex error transformations (common in prescribed performance controllers), to simultaneously impose constraints on the steady-state and transient response of the closed-loop, including maximum error, maximum overshoot, and minimum convergence rate his letter addresses the problem of robust finite-time tracking control with prescribed performance for robot manipulators experiencing uncertain inertia, external disturbance, and actuator fault. We develop a control strategy that incorporates the nonlinear H∞ concept into the backstepping approach, using a novel virtual control, to guarantee practical finite-time convergence to a trajectory, whilst the closed-loop L2 gain is less than a pre-specified value. We also use adaptive gains, instead of complex error transformations (common in prescribed performance controllers), to simultaneously impose constraints on the steady-state and transient response of the closed-loop, including maximum error, maximum overshoot, and minimum convergence rateT. The developed controller is not contingent on solving the Hamilton-Jacobi or Riccati equations and is free of the singularities associated with using fractional power in finite-time control. The performance and efficacy of the proposed control framework are demonstrated through simulation studies and comparisons with pertinent works.

To reliably localize and control wheeled autonomous rovers, their controllers must keep the wheels away from traction loss. In this paper, we develop a fast and practical traction control system for rovers that track dynamic trajectories on rough firm terrains, leveraging their normally existing redundant control directions. Trajectory-tracking performance is guaranteed by input-output linearizing a nonholonomic model of the system and employing an appropriate stabilizing control law. We propose a novel methodology to optimally lift the control signals at the rover’s output level to determine the control actions that enhance the system’s traction without affecting the tracking performance. The methodology uses the knowledge of wheels’ friction coefficients and estimation of normal and tractive forces based on a nonholonomic rover model to optimally distribute the tractive forces among the wheels. The novelty is in redefining the optimization problem in both lateral and longitudinal directions that require minimum information about wheel-ground interactions and leads to linear optimality conditions. We define the notion of total required force/moment at system’s center of mass to (i) introduce reference directions for tractive forces in the proposed cost functions, and (ii) identify the rover wheels fighting against the motion. To prevent wheel-fighting, we find sub-optimal solutions that suppress tractive forces at the fighting wheels. The proposed traction control system is implemented on a six-wheel autonomous Lunar rover and its efficacy is investigated by a developed software-in-the-loop simulation environment using Vortex Studio. This software simulates a 3-dimensional digital twin of the system, with different terrain and tire model options. When compared to the conventional pseudo-inverse solution, the developed traction controller demonstrates improved overall traction and it saves the rover from traction loss.

In the first part, this article presents an overview of Guidance, Navigation and Control (GNC) methodologies developed for space manipulators to perform in-orbit robotic missions, including but not limited to, on-orbit servicing, satellite/station assembly, probing extra-terrestrial objects and space debris mitigation. Some space mission concepts are briefly mentioned, for which space robotics is discussed to be among the most practical and universal solutions. Common phases of an in-orbit robotic mission are identified as: close-range rendezvous, attitude synchronization, target identification, manipulator deployment, capture, and if needed, post-capture maneuvers. Prominent GNC methodologies that are either proposed for or applicable to each phase are extensively reviewed. In the current article, the emphasis is placed on the study of GNC methodologies utilized in attitude synchronization, manipulator deployment, and capture phases, specially the ones reported for use in the two free-floating and free-flying operating regimes of space manipulators. Kinematics and dynamics of space manipulator systems are formulated to help unifying the presentation of the main ideas behind different GNC methodologies. Using a unified notation, comparison tables and discussions provided in this paper, researchers can compare various GNC approaches and contribute to the next-generation GNC systems for space robots. In addition, this survey aids technology users to learn about in-orbit robotic missions and choose appropriate GNC technologies for specific applications. In the second part of this paper, two families of emerging control schemes based upon reinforcement learning and geometric mechanics are introduced as promising research directions in the GNC of space robotic systems. The benefits of implementing these techniques to the GNC of in-orbit robotic missions are discussed. An exclusive study of environmental disturbances affecting space manipulators and their threat to long-term autonomy concludes this article.

An assessment of the dynamic performance of long combination vehicles (LCV) with steerable axles was undertaken to facilitate the regulation of LCVs for wider use on Canadian roads. A base dry box van A-train LCV and four steered combinations with different steerable axle mechanisms on the trailer are modelled using TruckSim. TruckSim's driver logic is augmented through an optimised controller to more accurately capture the driver's decision-making in response to the LCV dynamics. Anti-lock braking system (ABS) and traction control mechanisms are added to compensate for the reduced stability caused by improving the manoeuvrability. The models are run through highspeed lane change and turn simulations, the primary manoeuvres for the assessment of roll dynamics of LCVs. The configurations are compared in terms of standard performance parameters: static roll threshold, rearward amplification (RWA), load transfer ratio (LTR), highspeed off-tracking and transient off-tracking. All mechanisms are shown to satisfy standard stability requirements.

Similar to any service or product, industrialization of On-Orbit Servicing (OOS) demands performance enhancement through introducing relevant autonomy elements in planning and executing single and multiple servicing missions. This paper proposes an overall mission architecture for performing multiple on-orbit servicing missions by a fleet of servicers in the form of free-flying single-arm space manipulators. The architecture targets to improve the two key industrialization criteria of resource and service. In the far-range rendezvous with target satellites, the servicers burn most of their fuel. Furthermore, the time that servicers spend in transfer orbits determines the approximate duration of a servicing mission. Hence, as part of resource management, the presented architecture first identifies the main contributors to the fuel consumption and mission duration in far-range rendezvous phase of OOS missions being: (i) the location of the parking orbit, (ii) the type of transfer trajectories, and (iii) the dispatch scheduling. As the result, separate optimization loops are considered for minimizing the mission costs, across the OOS industry. Servicers are suggested to form an equally phased constellation in a parking orbit close to Sun-synchronous orbits in the Low Earth Orbital (LEO) region, where 57.5% of operational LEO satellites reside. A satellite in the parking orbit constellation is named “Administrator”, whose sole purpose is to plan and manage servicing missions. The Administrator determines the optimal number and sequence of servicing missions that must be performed by the available servicers, and the optimal transfer trajectories servicers shall follow to reach the targets. Upon completion of their missions, each servicer returns to the parking orbit and occupies the available position that requires the lowest fuel consumption to enter. In almost 90% of servicers' lifetime, they are in an idle state in the parking orbit awaiting dispatch or in transfer orbits. To enhance quality of the provided service, the proposed architecture suggests effective use of this time to task servicers with performing machine learning that helps improve the functionality of their guidance, navigation and control systems in upcoming missions. The task involves trajectory learning for a servicer's manipulator system in free-floating regime to reach a simulated moving target while avoiding (virtual) obstacles and compensating for environmental disturbances. Both supervised and unsupervised machine learning techniques are considered, and based on a qualitative analysis, the unsupervised DDPG algorithm is deemed most applicable in the free-floating trajectory learning task.

On-orbit servicing is a range of orbital services that comprises visual inspection, refueling, repairing, upgrading, assembly, and debris removal. It intends to increase the satellites' lifetime and enhance their performance. This paper proposes a methodology for designing smooth trajectories for long-range rendezvous of servicing satellites with moving targets, in on-orbit servicing missions. The methodology employs a multi-impulse shape-based trajectory planning algorithm for in-plane orbit transfer, based on the two-body problem. A multiobjective constrained optimization architecture is developed using a genetic algorithm to determine the optimal trajectories in the sense of Pareto optimality. Transfer time and control effort are considered as Pareto cost functions. Arriving at an orbiting target upon completion of the transfer and limitations in orbital elements are included as constraints. The latter constraint will help reduce the risk of collision in populated orbits by not entering those orbits, but maybe crossing them. The design variables are the orbital elements of the intermediate orbits, and the number and location of impulses. The location of the first impulse in the parking orbit indicates the waiting time before the commencement of orbit transfer that can impact the optimal solution when chasing a dynamic target. Compared to trajectory design methods using continuous thrust, the proposed technique has fewer design variables. The set of Pareto frontier solutions provides an on-orbit servicing mission with decision-making capabilities to choose a solution compatible with the priority requirements of the mission. The proposed methodology is specifically valid for the low Earth orbital regime (100–2000 Km altitude), where the gravitational field of the Earth dominates disturbances such as solar pressure, and the gravity of other celestial objects. To remain in this regime, constraints on intermediate orbits are imposed to ensure the altitude at apoapsides do not exceed 2000 km and at periapsides remains above 100 km. The performance of the resulting methodology is then compared with that of an optimal Lambert approach, in different case studies.

This paper outlines a concurrent design methodology for multidisciplinary systems, which employs tools of fuzzy theory for the tradeoff in the design space. This methodology enhances communication between designers from various disciplines through introducing the universal notion of satisfaction and expressing the behaviour of multidisciplinary systems using the notion of energy. It employs fuzzy rule-bases, membership functions and parametric connectives in fuzzy logic to formalize subjective aspects of design, resulting in a two-phase simplification of the multi-objective constrained optimization of a design process. The methodology attempts to find a pareto-optimal solution for the design problem. In the primary phase of the methodology, a fuzzy-logic model is utilized to identify a region in the design space that contains the pareto-optimal design state, and a proper initial state is suggested for the optimization in the secondary phase, where the pareto-optimal solution is found. Finally, the impact of the designer's subjective attitude on the design is adjusted based on a system performance by utilizing an energy-based model of multidisciplinary systems. As an application, it is shown that the design of a five-degree-of-freedom industrial robot manipulator can be enhanced by using the methodology.

This paper presents a two-step symplectic geometric approach to the reduction of Hamilton’s equation for open-chain, multi-body systems with multi-degree-of-freedom holonomic joints and constant momentum. First, symplectic reduction theorem is revisited for Hamiltonian systems on cotangent bundles. Then, we recall the notion of displacement subgroups, which is the class of multi-degree-of-freedom joints considered in this paper. We briefly study the kinematics of open-chain multi-body systems consisting of such joints. And, we show that the relative configuration manifold corresponding to the first joint is indeed a symmetry group for an open-chain multi-body system with multi-degree-of-freedom holonomic joints. Subsequently using symplectic reduction theorem at a non-zero momentum, we express Hamilton’s equation of such a system in the symplectic reduced manifold, which is identified by the cotangent bundle of a quotient manifold. The kinetic energy metric of multi-body systems is further studied, and some sufficient conditions are introduced, under which the kinetic energy metric is invariant under the action of a subgroup of the configuration manifold. As a result, the symplectic reduction procedure for open-chain, multi-body systems is extended to a two-step reduction process for the dynamical equations of such systems. Finally, we explicitly derive the reduced dynamical equations in the local coordinates for an example of a six-degree-of-freedom manipulator mounted on a spacecraft, to demonstrate the results of this paper.

This paper presents a unified geometric framework to input-output linearization of open-chain multi-body systems with symmetry in their reduced phase space. This leads us to an output tracking controller for a class of underactuated open-chain multi-body systems with holonomic and nonholonomic constraints. We consider the systems with multi-degree-of-freedom joints and possibly with non-zero constant total momentum (in the holonomic case). The examples of these systems are free-base space manipulators and mobile manipulators. We first formalize the control problem, and rigorously state an output tracking problem for such systems. Then, we introduce a geometrical definition of the end-effector pose and velocity error. The main contribution of this paper is reported in Section 5, where we solve for the input-output linearization of the highly nonlinear problem of coupled manipulator and base dynamics subject to holonomic and nonholonomic constraints. This enables us to design a coordinate-independent controller, similar to a proportional-derivative with feed-forward, for concurrently controlling a free-base, multi-body system. Finally, by defining a Lyapunov function, we prove in Theorem 3 that the closed-loop system is exponentially stable. A detailed case study concludes this paper.

This paper studies the geometry behind nonholonomic Hamilton's equation to present a two-stage reduction procedure for the dynamical equations of nonholonomic open-chain multi-body systems with multi-degree-of-freedom joints. In this process, we use the Chaplygin reduction and an almost symplectic reduction theorem. We first restate the Chaplygin reduction theorem on cotangent bundle for nonholonomic Hamiltonian mechanical systems with symmetry. Then, under some conditions we extend this theorem to include a second reduction stage using an extended version of the symplectic reduction theorem for almost symplectic manifolds. We briefly introduce the displacement subgroups and accordingly open-chain multi-body systems consisting of such joints. For a holonomic open-chain multi-body system, the relative configuration manifold corresponding to the first joint is a symmetry group. Hence, we focus on a class of nonholonomic distributions on the configuration manifold of an open-chain multi-body system that is invariant under the action of this group. As the first stage of reduction procedure, we perform the Chaplygin reduction for such systems. We then introduce a number of sufficient conditions for a reduced system to admit more symmetry due to the action of the relative configuration manifolds of other joints. Under these conditions, we present the second stage of the reduction process for nonholonomic open-chain multi-body systems with multi-degree-of-freedom joints. Finally, we explicitly derive the reduced dynamical equations in the local coordinates for an example of a two degree-of-freedom crane mounted on a four-wheel car to illustrate the results of this paper.

This paper presents a generalized exponential formula for Forward and Differential Kinematics of open-chain multi-body systems with multi-degree-of-freedom, holonomic and nonholonomic joints. The notion of lower kinematic pair is revisited, and it is shown that the relative configuration manifolds of such joints are indeed Lie groups. Displacement subgroups, which correspond to different types of joints, are categorized accordingly, and it is proven that except for one class of displacement subgroups the exponential map is surjective. Screw joint parameters are defined to parameterize the relative configuration manifolds of displacement subgroups using the exponential map of Lie groups. For nonholonomic constraints the admissible screw joint speeds are introduced, and the Jacobian of the open-chain multi-body system is modified accordingly. Computational aspects of the developed formulation for Forward and Differential Kinematics of open-chain multi-body systems are explored by assigning coordinate frames to the initial configuration of the multi-body system, employing the matrix representation of SE(3) and choosing a basis for se(3). Finally, an example of a mobile manipulator mounted on a spacecraft, i.e., a six-degree-of-freedom moving base, elaborates the computational aspects.

This article details a holistic concurrent design framework, based on fuzzy logic, which is suitable for multidisciplinary systems. The methodology attempts to enhance communication and collaboration between different disciplines through introducing the universal notion of satisfaction and expressing the holistic behavior of multidisciplinary systems using the notion of energy. Throughout the design process, it uses fuzzy logic to formalize subjective aspects of design including the impact of the designer’s attitude, resulting in the simplification of the multi-objective constrained optimization process. In the final phase, the methodology adjusts the designer’s subjective attitude based on a holistic system performance by utilizing an energy-based model of multidisciplinary systems. The efficiency of the resulting design framework is illustrated by improving the design of a 5-degree-of-freedom industrial robot manipulator.

This paper discusses a practical approach to the concurrent design of robot manipulators, which is based on an alternative design methodology, namely Holistic Concurrent Design (HCD), as well as the utilization of a modular hardware-in-the-loop simulation. Holistic concurrent design is a systematic design methodology for mechatronic systems that formalizes subjective notions of design, resulting in the simplification of the multi-objective constrained optimization process. Its premise is to enhance the communication between designers with various backgrounds and customers, and to consider numerous design variables with different natures concurrently. The methodology redefines the ultimate goal of design based on the qualitative notion of satisfaction, and formalizes the effect of designer’s subjective attitude in the process. The hardware-in-the-loop platform involves physical joint modules and the control unit of a manipulator in addition to the software simulation to reduce modeling complexities and to take into account physical phenomena that are hard to be captured mathematically. This platform is implemented in the HCD design architecture to reliably evaluate the design attributes and performance supercriterion during the design process. The resulting architecture is applied to redesigning kinematic, dynamic and control parameters of an industrial manipulator.

This paper presents a geometrical approach to the dynamical reduction of a class of constrained mechanical systems. The mechanical systems considered are with affine nonholonomic constraints plus a symmetry group. The dynamical equations are formulated in a Hamiltonian formalism using the Hamilton–d'Alembert equation, and constraint forces determine an affine distribution on the configuration manifold. The proposed reduction approach consists of three main steps: (1) restricting to the constrained submanifold of the phase space, (2) quotienting the constrained submanifold, and (3) identifying the quotient manifold with a cotangent bundle. Finally, as a case study, the dynamical reduction of a two-wheeled rover on a rotating disk is detailed. The symmetry group for this example is the relative configuration manifold of the rover with respect to the inertial space. The proposed approach in this paper unifies the existing reduction procedures for symmetric Hamiltonian systems with conserved momentum, and for Chaplygin systems, which are normally treated separately in the literature. Another characteristic of this approach is that although it tracks the structure of the equations in each reduction step, it does not insist on preserving the properties of the system. For example, the resulting dynamical equations may no longer correspond to a Hamiltonian system. As a result, the invariance condition of the Hamiltonian under a group action that lies at the heart of almost every reduction procedure is relaxed.

This paper presents the dynamical reduction and feedback linearization of the Lunar Exploration Light Rover to design an output-tracking torque controller for such a system. The rover is dynamically modeled as a system of rigid bodies that could be considered as a symmetric mechanical system. Its dynamical equations are reduced using the Chaplygin reduction of nonholonomic systems. The reduced dynamical system only involves the rotation and steering angles of the port front wheel. The steering angle and the travelled distance of the vehicle are considered the outputs of the resulting control system. Feedback linearization in the reduced phase space is then employed to derive a proportional-integral-derivative feedback, feedforward torque controller for the rover. Finally, a series of simulation studies are provided to show the robustness of the resulting controller under uncertainties of the dynamical parameters and the disturbance force in the form of friction at the rear wheels.

This paper outlines an alternative modeling scheme for mechatronic systems, as a basis for their concurrent design. The approach divides a mechatronic system into three generic subsystems, namely generalized executive, sensory and control, and links them together utilizing a combination of bond graphs and block diagrams. It considers the underlying principles of a multidisciplinary system, and studies the flow of energy and information throughout its different constituents. The first and second laws of thermodynamics are reformulated for mechatronic systems, and as a result three holistic design criteria, namely energy, entropy and agility, are defined. These criteria are formulated using the bond graph representation of a mechatronic system. As a case study, the three criteria are employed separately for concurrent design of a five degree-of-freedom industrial robot manipulator.

The elastica is a curve in $\R^3$ that is stationary under variations of the integral of the square of the curvature. Elastica is viewed as a dynamical system that arises from the second order calculus of variations, and its quantization is discussed., Comment: 55 pages

Robot manipulators are good examples of complex engineering systems, where designers occasionally employ a subsystem-partitioning approach for their analysis and synthesis. The design methodology is traditionally based on the sequential decomposition of mechanical, electromechanical, and control/instrumentation subsystems, so that at each step a subset of design variables is considered separately (Castano et al., 2002). Although conventional decoupled or loosely-coupled approaches of design seem intuitively practical, they undermine the interconnection between various subsystems that may indeed play a crucial role in multidisciplinary systems. The necessity of communication and collaboration between the subsystems implies that such systems ought to be synthesized concurrently. In the concurrent design process, design knowledge is accumulated from all the participating disciplines, and they are offered equal opportunities to contribute to each state of design in parallel. The synergy resulting from integrating different disciplines in concurrent design has been documented in several case studies, to the effect that the outcome is a new and previously unattainable set of performance characteristics (Hewit, 1996). However, the challenge in a concurrent design process is that the multidisciplinary system model can become prohibitively complicated; hence computationally demanding. Plus, a large number of multidisciplinary objective and constraint functions must be taken into account, simultaneously, with a great number of design variables. As the complexity of the system model increases, in terms of the interactions between various subsystems, the coordination of all the constraints distributed in different disciplines becomes more difficult, in order to maintain the consistency between performance specifications and design variables. Within the context of robotics, several ad hoc techniques of concurrent engineering have been reported in the literature. They are innovative design schemes for specific systems, such as Metamorphic Robotic System (Chirikjian, 1994), Molecule (Rus & McGray, 1998), Miniaturised Self-Reconfigurable System (Yoshida et al., 1999), Crystalline (Rus & Vona, 2000), and Semi-Cylindrical Reconfigurable Robot (Murata et al., 2000). But, more systematic approaches have been suggested by other researchers beyond the robotics community to tackle the challenge of high dimensionality in concurrent design. These approaches can be divided into two major groups. The first group translates the model complexity into a large volume of computations, and then attempts to find efficient algorithms or parallel 12