# Kinematics of Tree-Type Robotic Systems

01 Jan 2013-Vol. 62, pp 57-72

TL;DR: Using the present derivation, link-to-link velocity transformation turns out to be a special case of the module- to-module velocity transformation (Shah et al. 2012a) presented in this chapter.

Abstract: Kinematic modeling of a tree-type robotic system is presented in this chapter. In order to obtain kinematic constraints, a tree-type topology is first divided into a set of modules. The kinematic constraints are then obtained between these modules by introducing the concepts of module-twist, module-joint-rate, etc. This helps in obtaining the generic form of the Decoupled Natural Orthogonal Complement (DeNOC) matrices for a tree-type system with the help of module-to-module velocity transformations. Using the present derivation, link-to-link velocity transformation (Saha 1999a, b) turns out to be a special case of the module-to-module velocity transformation (Shah et al. 2012a) presented in this chapter.

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TL;DR: The main purpose of this work is to model a non holonomic mobile platform and to develop a hybrid algorithm for avoiding dynamic obstacles.

Abstract:
Purpose
Most of the conventional humanoid modeling approaches are not successful in coupling different branches of the tree-type humanoid robot. In this paper, a tree-type upper body humanoid robot with mobile base is modeled. The main purpose of this work is to model a non holonomic mobile platform and to develop a hybrid algorithm for avoiding dynamic obstacles. Decoupled Natural Orthogonal Complement methodology effectively combines different branches of the humanoid body during dynamic analysis. Collision avoidance also plays an important role along with modeling methods for successful operation of the upper body wheeled humanoid robot during real-time operations. The majority of path planning algorithms is facing problems in avoiding dynamic obstacles during real-time operations. Hence, a multi-fusion approach using a hybrid algorithm for avoiding dynamic obstacles in real time is introduced.
Design/methodology/approach
The kinematic and dynamic modeling of a humanoid robot with mobile platform is done using screw theory approach and Newton–Euler formulations, respectively. Dynamic obstacle avoidance using a novel hybrid algorithm is carried out and implemented in real time. D star lite and a geometric-based hybrid algorithms are combined to generate the optimized path for avoiding the dynamic obstacles. A weighting factor is added to the D star lite variant to optimize the basic version of D star lite algorithm. Lazy probabilistic road map (PRM) technique is used for creating nodes in configuration space. The dynamic obstacle avoidance is experimentally validated to achieve the optimum path.
Findings
The path obtained using the hybrid algorithm for avoiding dynamic obstacles is optimum. Path length, computational time, number of expanded nodes are analysed for determining the optimality of the path. The weighting function introduced along with the D star lite algorithm decreases computational time by decreasing the number of expanding nodes during path generation. Lazy evaluation technique followed in Lazy PRM algorithm reduces computational time for generating nodes and local paths.
Originality/value
Modeling of a tree-type humanoid robot along with the mobile platform is combinedly developed for the determination of the kinematic and dynamic equations. This paper also aims to develop a novel hybrid algorithm for avoiding collision with dynamic obstacles with minimal computational effort in real-time operations.

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TL;DR: In this paper, the Euler-Lagrange independent equations of motion of serial multibody systems consisting of rigid bodies in a serial kinematic chain are derived with the aid of the decoupled natural orthogonal complement matrices associated with the velocity constraints of the connecting bodies.

Abstract: Constrained dynamic equations of motion of serial multibody systems consisting of rigid bodies in a serial kinematic chain are derived in this paper. First, the Newton-Euler equations of motion of the decoupled rigid bodies of the system at hand are written. Then, with the aid of the decoupled natural orthogonal complement (DeNOC) matrices associated with the velocity constraints of the connecting bodies, the Euler-Lagrange independent equations of motion are derived. The De NOC is essentially the decoupled form of the natural orthogonal complement (NOC) matrix, introduced elsewhere. Whereas the use of the latter provides recursive order n-n being the degrees-of-freedom of the system at hand-inverse dynamics and order n 3 forward dynamics algorithms, respectively, the former leads to recursive order n algorithms for both the cases. The order n algorithms are desirable not only for their computational efficiency but also for their numerical stability, particularly, in forward dynamics and simulation, where the system's accelerations are solved from the dynamic equations of motion and subsequently integrated numerically. The algorithms are illustrated with a three-link three-degrees-of-freedom planar manipulator and a six-degrees-of-freedom Stanford arm.

103 citations

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01 Apr 1997TL;DR: An analytical approach is suggested in this paper based on the symbolic Gaussian elimination of the inertia matrix that reveal recursive relations among the elements of the resulting matrices that can be done with the complexity of order n, O(n).

Abstract: A decomposition of the manipulator inertia matrix is essential, for example, in forward dynamics, where the joint accelerations are solved from the dynamical equations of motion. To do this, unlike a numerical algorithm, an analytical approach is suggested in this paper. The approach is based on the symbolic Gaussian elimination of the inertia matrix that reveal recursive relations among the elements of the resulting matrices. As a result, the decomposition can be done with the complexity of order n, O(n), where n being the degrees of freedom of the manipulator, as opposed to an O(n/sup 3/) scheme, required in the numerical approach. In turn, O(n) inverse and forward dynamics algorithms can be developed. As an illustration, an O(n) forward dynamics algorithm is presented.

71 citations

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TL;DR: The derivation allows one to write a recursiveforward-dynamics algorithm for Forsimulation purposes whose computational complexity is of order a, i.e., O(n) n being the degrees offreedom of the robot under study.

Abstract: This paper presents the analytical derivation of the inertia matrix and its inverse for an open-loop, serial-chain mbot. The derivation allows one to write a recursiveforward-dynamics algorithmforsimulation purposes whose computational complexity is of order a, i.e., O(n) n being the degrees offreedom of the robot under study The proposed methodology is based on the Gaussian elimination of the inertia matrix, in contrast to, say, Kalmanftltering; which is proposed elsewhere. The derivation is illustrated with a three-degrees-of-freedom planar robot.

37 citations

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TL;DR: In this paper, a legged robot is modeled as a floating-base tree-type system where the foot-ground interactions are represented as external forces and moments, and recursive algorithms for inverse and forward dynamics are proposed by using inter-and intra-modular recursions for the first time.

Abstract: In this paper, a legged robot is modeled as a floating-base tree-type system where the foot-ground interactions are represented as external forces and moments. Dynamic formulation thus obtained is independent of the configuration or state of the legged robot. Framework for dynamic modeling is proposed with the concept of kinematic modules, where each module is a set of serially connected links. Legged robots are then considered to have several kinematic modules, and kinematic constraints among these modules are obtained in a similar way as those between the links. The latter approach turns out to be a special case of the former where each module has only one link. A velocity transformation based approach is used to obtain the minimal-order equations of motion, and module-level analytical expressions for the vectors and matrices appearing in them. Recursive algorithms for inverse and forward dynamics are proposed by using inter- and intra-modular recursions for the first time. Analyses of a planar biped and spatial quadruped are presented using the proposed methodology. Effectiveness of the proposed algorithms to model-based control schemes is also provided.

36 citations

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TL;DR: In this article, a new constraint wrench formulation for closed-loop systems is presented using two-level recursions, namely, subsystem level and body level, where a subsystem is referred here as the serial- or tree-type branches of a spanning tree obtained by cutting the appropriate joints of the closed loops of the system at hand.

Abstract: In order to compute the constraint moments and forces, together referred here as wrenches, in closed-loop mechanical systems, it is necessary to formulate a dynamics problem in a suitable manner so that the wrenches can be computed efficiently. A new constraint wrench formulation for closed-loop systems is presented in this paper using two-level recursions, namely, subsystem level and body level. A subsystem is referred here as the serial- or tree-type branches of a spanning tree obtained by cutting the appropriate joints of the closed loops of the system at hand. For each subsystem, unconstrained Newton-Euler equations of motion are systematically reduced to a minimal set in terms of the Lagrange multipliers representing the constraint wrenches at the cut joints and the driving torques/forces provided by the actuators. The set of unknown Lagrange multipliers and the driving torques/forces associated to all subsystems are solved in a recursive fashion using the concepts of a determinate subsystem. Next, the constraint forces and moments at the uncut joints of each subsystem are calculated recursively from one body to another. Effectiveness of the proposed algorithm is illustrated using a multiloop planar carpet scraping machine and the spatial RSSR (where R and S stand for revolute and spherical, respectively) mechanism.

31 citations