Proceedings of the IMA Conference on Mathematics of Robotics

A. Costa and M. Farber

Full Paper:
https://doi.org/10.19124/ima.2015.001.01

Manifolds and simplicial complexes traditionally appear in robotics as configuration spaces of mechanical systems. The classical approach becomes inadequate if we are dealing with a large system, i.e. with a system depending on a large number of metric parameters, since these parameters cannot be measured without errors and small errors make significant impact on the structure of the obtained space. A more realistic approach for modelling large systems is based on the assumption that their configuration spaces are random with properties described using the language of probability theory.

S. Bhattacharya and R. Ghrist

Full Paper
https://doi.org/10.19124/ima.2015.001.02

We consider the problem of optimal path planning in different homotopy classes in a given environment. Though important in robotics applications, path-planning with reasoning about homotopy classes of trajectories has typically focused on subsets of the Euclidean plane in the robotics literature. The problem of finding optimal trajectories in different homotopy classes in more general configuration spaces (or even characterizing the homotopy classes of such trajectories) can be difficult. In this paper we propose automated solutions to this problem in several general classes of configuration spaces by constructing presentations of fundamental groups and giving algorithms for solving the word problem in such groups. We present explicit results that apply to knot and link complements in 3-space, and also discuss how to extend to cylindrically deleted coordination spaces of arbitrary dimension.

D. Blanc and N. Shvalb

Full Paper
https://doi.org/10.19124/ima.2015.001.03

Most useful linkages are spatial. Commonly, for such a mechanism there are configurations c 2 C were two links or more intersect. We introduce a mathematical framework to deal with the neighbourhood of c and define virtual configurations in which two links touch. Cauchy sequences are then used to approximate these configurations. As for the global structure, we show how the completion of the configuration space relates to a virtual configuration space were all lines meet the origin. We conclude the paper with some explicit example.

K. Kalyanam and M. Pachter

Full Paper
https://doi.org/10.19124/ima.2015.001.04

The optimal control of a “blind” pursuer searching for an evader moving on a road network toward a set of goal vertices is considered. The evader’s speed is time-varying and not known to the pursuer; although upper and lower bounds on the speed are assumed known. To aid the pursuer, certain roads in the network have been instrumented with Unattended Ground Sensors (UGSs) that detect the evader’s passage. When the pursuer arrives at an instrumented node, the UGS therein informs the pursuer if and when the evader visited the node. The pursuer’s motion is not restricted to the road network. In addition, the pursuer can choose to wait/loiter at any UGS location/node. At time 0, the evader passes by an entry node and proceeds towards one of the exit nodes. The pursuer also arrives at this entry node after some delay and is thus informed about the presence of the intruder/evader in the network, whereupon the chase is on. Capture entails the pursuer and evader being co-located at an UGS location. If this happens, the UGS is triggered and this information is instantaneously relayed to the pursuer, thereby enabling capture. On the other hand, if the evader reaches one of the exit nodes without being captured, he is deemed to have escaped. We provide an algorithm that computes the maximum initial delay at the entry node for which capture is guaranteed and the corresponding optimal pursuit policy.

C.S. Borcea and I. Streinu

Full Paper
https://doi.org/10.19124/ima.2015.001.05

A one-parameter deformation of a periodic bar-and-joint framework is expansive when all distances between joints increase or stay the same. In dimension two, expansive behavior can be fully explained through our theory of periodic pseudo-triangulations. However, higher dimensions present new challenges. In this paper we study a number of periodic frameworks with expansive capabilities in dimension d _ 3 and register both similarities and contrasts with the two-dimensional case.

M.A. Post

Full Paper
https://doi.org/10.19124/ima.2015.001.06

A wide variety of approaches exist for dealing with uncertainty in robotic reasoning, but relatively few consider the propagation of statistical information throughout an entire robotic system. The concept of Bayesian Robot Programming involves making decisions based on inference into probability distributions, but can be complex and difficult to implement due to the number of priors and random variables involved. In this work, we apply Bayesian network structures to a modified Bayesian programming paradigm to provide intuitive structure and simplify the programming process. The use of discrete random variables in the network can allow high inference speeds, and an efficient programming toolkit suitable for use on embedded platforms has been developed for use on mobile robots. A simple example of navigational reasoning for a small mobile robot is provided as an example of how such a network can be used for decisional programming.

O.Roussel, M. Renaud and M. Taïx

Full Paper
https://doi.org/10.19124/ima.2015.001.07

In this paper, we address the problem of inverse geometry for a Kirchhoff elastic rod at equilibrium manipulated at its ends by robotic grippers. Based on the explicit formulation of extremal curves in terms of elliptic functions, we derive closed forms for rod shape and sensitivity in the planar case and we show how this can be used efficiently in robotics applications. More specifically, a numerical optimization approach is presented to solve the inverse geometry problem in the general 3-D case and applied to the planar case using closed forms previously introduced.

J. Carpentier, A. Del Prete, N. Mansard and J. Laumond

Full Paper
https://doi.org/10.19124/ima.2015.001.08

We propose an original formulation to describe the motion of a rolling object in contact with a at surface, and propose a complete application of this model to generate optimal walk of a humanoid system. We derive an analytic formulation of the contact equations which does not require any numerical approximation. We then show how the model performs when applied to the simulation of bipedal locomotion: by replacing polyhedral models of the feet by ellipsoidal ones, the numerical algorithms run faster and provide higher quality results. Furthermore, as any convex shape may be approximated locally by an ellipsoid, the scope of the model goes beyond locomotion simulation.

C. Linton, W. Holderbaum, J. Biggs

Full Paper
https://doi.org/10.19124/ima.2015.001.09

This paper proposes several extensions to the existing method of parametrizing speed along a prescribed path. The velocity is modified by isotropically stretching/shrinking the tangent space. The path in closed form is determined by substitution, without the computational cost of re-integrating the velocity function. This concept can be extended to anisotropic stretching/shrinking of the momentum space to change the direction and magnitude of the momentum vector.

The physical constraints of the actuators on increasing and decreasing momentum (and its differentials) are incorporated into a parametrization function that achieves maximum distance for a given input of energy and satisfies boundary conditions on the momentum.

This method of time parametrization especially applies to Geometric Control, where the Hamiltonian minimizes some cost function and matches the boundary configuration constraints but not the velocity constraints. The optimal (geometric) trajectory is modified by the parametrization so that the cost function is minimized if the stretching is stopped at any time. The forces stretching the momentum space are identifiable from the formulation.

An asymmetric rigid satellite illustrates the modification of angular momentum, with independent parametrizations of the linear momentum.

J. Ratajczak and K. Tchoń

Full Paper
https://doi.org/10.19124/ima.2015.001.10

We propose a transfer of the concept of dynamically consistent Jacobian inverse from robotic manipulators to non-holonomic robotic systems. This transfer exploits an analogy between these two classes of systems established within the endogenous configuration space approach. Similarly as for robotic manipulators, the dynamic consistency of the Jacobian inverse for non-holonomic systems distinguishes itself by two features: it does not transmit some forces from the configuration space to the operational space, and it defines a decoupling of forces acting in the configuration space into a component coming from the operational space and an internal component affecting only the motion in the configuration space. The dynamically consistent Jacobian inverse is used to solve a motion planning problem for the rolling ball.

K. Tchoń and I. Góral

Full Paper
https://doi.org/10.19124/ima.2015.001.11

This paper is devoted to the motion planning problem of nonholonomic robotic systems whose kinematics are represented by a driftless control system with outputs. The motion planning problem is defined in terms of the inversion of the end-point map of the system, that may be accomplished by a Jacobian algorithm. The Lagrangian Jacobian inverse is introduced and examined, and a corresponding Lagrangian Jacobian motion planning algorithm is designed. Performance of this algorithm is illustrated by solving a motion planning problem for the rolling ball.

P.Zech, H. Xiong and J. Piater

Full Paper
https://doi.org/10.19124/ima.2015.001.12

In this paper we consider black-box optimization of objects’ grasp density functions relative to a gripper’s orientation. For this, we introduce Monte-Carlo Tree Search on the unit quaternion manifold. Our experimental evaluation shows that our method is feasible and allows finding grasps of arbitrary quality with regard to a fixed budget of objective function evaluations.

J.D. Biggs

Full Paper
https://doi.org/10.19124/ima.2015.001.13

This paper presents a new quaternion-based attitude tracking controller. A general Lyapunov function is defined whose derivative is control dependent and a control is chosen to guarantee asymptotic stability of the zero-error state. The corresponding closed-loop error dynamics are shown to reduce to a simple 1 degree of freedom description in terms of the eigen-axis angle error. The main contribution of this paper is to present a special case where the closed-loop error dynamics reduce to a simple linear oscillator description (without the need for linearisation). This means that the controller can be tuned to guarantee exponentially fast tracking with a damped response and without oscillation.

T. Lee

Full Paper
https://doi.org/10.19124/ima.2015.001.14

This paper is focused on tracking control for a rigid body payload, that is connected to an arbitrary number of quadrotor unmanned aerial vehicles via rigid links. A geometric adaptive controller is constructed such that the payload asymptotically follows a given desired trajectory for its position and attitude in the presence of uncertainties. The coupled dynamics between the rigid body payload, links, and quadrotors are explicitly incorporated into control system design and stability analysis. These are developed directly on the nonlinear configuration manifold in a coordinate-free fashion to avoid singularities and complexities that are associated with local parameterizationsT.

J. Jamieson and J.D. Biggs

Full Paper
https://doi.org/10.19124/ima.2015.001.15

This paper presents a kinematic path planning method for a slender autonomous underwater vehicle (AUV) using a technique based on the Rapid-Exploring Random Tree (RRT) algorithm extended to the Euclidean group of motions SE(3). The paths are generated by connecting sub-Riemannian curves on SE(3) that are optimal with respect to a weighted quadratic cost function of the translational and angular velocities subject to the kinematic constraints of the vehicle and are C1 smooth for translational motions. Unlike the majority of other AUV path planning methods, our framework provides a full description of the rotational and translational position as a function of time and the curves satisfy the differential constraints of the vehicle. The modified RRT method is demonstrated in two scenarios, the first with a sparse obstacle field and the second with a dense obstacle field. In both cases the algorithm successfully generates a six degree-of-freedom path from the initial position to a desired final position.

T.-D. Rad and H.-P. Schröcker

Full Paper
https://doi.org/10.19124/ima.2015.001.16

We characterise the kinematic image of the constraint variety of a 2R dyad as a regular ruled quadric in a three-space that contains a “null quadrilateral”. Three prescribed poses determine, in general, two such quadrics. This allows us to modify a recent algorithm for the synthesis of 6R linkages in such a way that two consecutive revolute axes coincide, thus producing a 5R linkage. Using the classical geometry of twisted cubics on a quadric, we explain some of the peculiar properties of the resulting synthesis procedure for 5R linkages.

G. De La Torre and E. A. Theodorou

Full Paper
https://doi.org/10.19124/ima.2015.001.17

In this paper we present a stochastic variational integrator and its linearization. In order to motivate the use of the proposed variational integrator the Stochastic Differential Dynamical Programming (S-DDP) algorithm is considered as a benchmark for comparison. Specifically, we are interested in investigating if it is advantageous to utilize the variational integrator to propagate system trajectories and linearize system dynamics. Through numerical experiments we show that the Stochastic Differential Dynamical Programming algorithm becomes less dependent on the discretization time step and more predictable when it utilizes the proposed integrator. Furthermore, we show that a significant reduction in computational time can be achieved without sacrificing algorithm performance. Therefore, the proposed variational integrator can be used to enable real-time implementation of nonlinear optimal control algorithms.

M. M. Ghazaei Ardakani, A. Robertsson, R. Johansson

Full Paper
https://doi.org/10.19124/ima.2015.001.18

Robotic trajectory generation is reformulated as a controller design problem. For minimum-jerk trajectories, an optimal controller using the Hamilton-Jacobi-Bellman equation is derived. The controller instantaneously updates the trajectory in a closed-loop system as a result of changes in the reference signal. The resulting trajectories coincide with piece-wise fifth-order polynomial trajectories for piece-wise constant target states. Since having hard constraints on the final time poses certain robustness issues, a smooth transition between the finite-horizon and an infinite-horizon problem is developed. This enables to switch softly to a tracking mode when a moving target is reached.

T. Lee, M. Leoky and N. Harris McClamroch

Full Paper
https://doi.org/10.19124/ima.2015.001.19

This paper provides global formulations of Lagrangian and Hamiltonian variational dynamics evolving on a manifold embedded in Rn, which appears often in robotics and multi body dynamics. Euler–Lagrange equations and Hamilton’s equations are developed in a coordinate-free fashion on a manifold, without relying on local parameterizations that may lead to singularities and cumbersome equations of motion. The proposed intrinsic formulations of Lagrangian and Hamiltonian dynamics are expressed compactly, and they are useful in analysis and computation of the global dynamics. These are illustrated by dynamic systems on the unit-spheres and the special orthogonal group.

B. Bongardt and F. Kirchner

Full Paper
https://doi.org/10.19124/ima.2015.001.20

For the computation of rigid body dynamics, the Newton-Euler equations represent a crucial relation unifying the laws of motion by Newton and Euler using the language of instantaneous screws. Typically, Newton-Euler equations are stated in spatial or in body coordinates, respectively. In this paper, a general formulation of Newton-Euler equations is provided for arbitrary reference systems. In particular, the general form unifies the known equations in spatial and body coordinates. To the best of the authors’ knowledge, this relation between the spatial and the body form has not been reported in literature. The novel formulation is based on the concept of time differentiation with respect to moving reference systems.

J.M. Selig

Full Paper
https://doi.org/10.19124/ima.2015.001.21

Three different concepts from the past are reviewed from a more modern standpoint. Constructing an equimomental system of four point-masses to an arbitrary rigid-body; the conditions for the generalised mass matrix of a serial robot to be constant and how to dynamically balance an arbitrary rigid body so that it is symmetrical about a given axis. Connections are made to the geometry of a 3-dimensional Veronese variety.

C. Deeks

Full Paper
https://doi.org/10.19124/ima.2015.001.22

This paper describes a novel approach to the control of a number of autonomous robots operating concurrently but without direct communication or coordination between them. Each automaton is a part of each of the others’ dynamic local environments, and control actions need to be selected in response to observations of this local environment. A variation on Qlearning is introduced where the set of actions available to an automaton can be artificially constrained in response to observations and so with learned policies being recalculated online. The model to which this approach has been applied is a representation of a set of production line automata, where there may be a desire for a greater degree of concurrent operation to make the whole production line more efficient, but where there is also the desire to avoid long ramp-up or configuration times. It is shown that robust behaviour within set geometric constraints can be observed, even when the problem is set so that the automata cannot avoid obstructing each other. Training not just one automaton with a machine learning technique, but training a family of them concurrently, is therefore shown to be a feasible candidate for more widespread application in coordinated control tasks in dynamic environments.

M. G. Weiß

Full Paper
https://doi.org/10.19124/ima.2015.001.23

We describe a class of 6R robots for which we can give explicit formulas for the 16 solutions in the special case of the tool orientation in parallel with axes of the base coordinate system. This setting can serve as the starting point for research towards better understanding of the 16 solutions.

J. Brinker, M. Lorenz, S.C. Eddine and B. Corves

Full Paper
https://doi.org/10.19124/ima.2015.001.24

This paper describes two major aspects within the research fields of robotics: First, the analytical derivation of the Jacobian matrix of mechanisms with complex kinematics and second, the performance evaluation of redundantly actuated parallel kinematic manipulators with different types of actuators. For mechanisms with simple kinematics, the Jacobian matrix can usually be obtained by straightforward derivation and differentiation of the closure equation. Complex architectures, however, require comprehensive analytical approaches. Accordingly, an existing method to determine the Jacobian based on selection matrices is applied to a 3-RPRRR mechanism. In order to characterize and evaluate corresponding structures, the concept of power manipulability ellipsoids is applied. Thus, the common problem of the Jacobian’s inhomogeneity due to a mixed translational and/or rotational drive configuration is avoided.

R. C. Harwin, P.M. Sharkey, W. S. Harwin, S. M. Scott, D. M. Hawkins and S. P. Collins

Full Paper
https://doi.org/10.19124/ima.2015.001.25

This paper examines the kinematics of parallel linkage arrangement referred to as a tripod. The mechanism has 6 degrees of freedom and is isomorphic to a 6-3 Stewart platform. A practical and fast forward kinematics solution is given, based on mapping to the 6-3 Stewart platform, and has been implemented on a prototype platform in both Python and Matlab. The platform is designed for a set of Kirkpatrick-Baez X-ray mirrors and the results show that an accuracy of the order of tens of nanometres is possible.

A. Müller, D. Zlatanov

Full Paper
https://doi.org/10.19124/ima.2015.001.26

The paper revisits the definition and the identification of the singularities of kinematic chains and mechanisms. The degeneracy of the kinematics of an articulated system of rigid bodies cannot always be identified with the singularities of the configuration space. Local analysis can help identify kinematic chain singularities and better understand the way the motion characteristics change at such configurations. An example is shown that exhibits a kinematic singularity although its configuration space is a smooth manifold

E. Rimon and G. Mano

Full Paper
https://doi.org/10.19124/ima.2015.001.27

This paper considers the synthesis of time optimal paths for a mobile robot docking at targets located on the boundary of a polygonal obstacle in IR2. The robot must avoid the obstacle as well as satisfy velocity dependent braking safety constraints during the docking process. The classical Brachistochrone problem studies the time optimal path of a particle moving in an obstacle free environment subject to a constant force field. By encoding the braking safety constraint as a force field surrounding the obstacle, the paper generalizes the Brachistochrone problem into synthesis of time optimal paths for a mobile robot attempting to dock against an obstacle boundary. To solve the time optimal docking problem, the safe travel time functional, a path dependent function, is formulated. Convexity properties of this functional allow computation of the time optimal docking path as a convex optimization problem in O(n3 log(1/ϵ)) time, where n is the number of obstacle vertices and ϵ is the desired solution accuracy.

S. Battle

Full Paper
https://doi.org/10.19124/ima.2015.001.28

In the foundational cybernetics text, Design for a Brain, W. Ross Ashby introduces an adaptive system called the homeostat, and speculates about the possibility of creating a mobile homeostat “with its critical states set so that it seeks situations of high illumination.” Simulations demonstrate the viability of using the classic homeostat architecture to control a mobile robot demonstrating ultrastability in adapting to an environment where the goal is to stay within range of a single source of illumination. This paper explores a novel physical embodiment of Ashby’s classic homeostat in a mobile robot with three degrees of freedom (2 translational, 1 rotational). The hypothesis, borne out by tests, is that the topological configuration of the robot as determined by simulation, will carry over into the physical robot.