Improvements on formulations of dynamic models for robotic systems

In this work, a rigorous framework is developed for the modeling and control of spherical robotic vehicles. Motivation for this work stems from the development of Moball, which is a self-propelled sensor platform that harvests kinetic energy from local wind fields. To study Moball's dynamics, the processes of Lagrangian reduction and reconstruction are extended to robotic systems with symmetry-breaking potential energies, in order to simplify the resulting dynamic equations and expose mathematical structures that play an important role in subsequent control-theoretic tasks. These results apply to robotic systems beyond spherical robots. A formulaic procedure is introduced to derive the reduced equations of motion of most spherical robots from inspection of the Lagrangian. This adaptable procedure is applied to a diverse set of robotic systems, including multirotor aerial vehicles. Small time local controllability (STLC) results are derived for barycentric spherical robots (BSR), which are spherical vehicles whose locomotion depends on actuating the vehicle's center of mass (COM) location. STLC theorems are introduced for an arbitrary BSR on flat, sloped, or smooth terrain. I show that STLC depends on the surjectivity of a simple steering matrix. An STLC theorem is also derived for a class of commonly encountered multirotor vehicles. Feedback linearizing and PID controllers are proposed to stabilize an arbitrary spherical robot to a desired trajectory over smooth terrain, and direct collocation is used to develop a feedforward controller for Moball specifically. Moball's COM is manipulated by a novel system of magnets and solenoids, which are actuated by a ballistic-impulse controller that is also presented. Lastly, a motion planner is developed for energy-harvesting vehicles. This planner charts a path over smooth terrain while balancing the desire to achieve scientific objectives, avoid hazards, and the imperative of exposing the vehicle to environmental sources of energy such as local wind fields and topology. Moball's design details and experimental results establishing Moball's energy-harvesting performance (7W while rolling at a speed of 2 m/s), are contained in an Appendix.

: The center of mass (COM) of a body or segment is the point about which the mass of the body or segment is evenly distributed. Moment of inertia (MOI) is the measure of a segment or object's resistance to changes in angular velocity. Mass properties, such as COM, MOI, and mass, allow for characterization of objects and for an easy comparison of the effects of force and torque on the dynamics of the object. This report describes and demonstrates the use of a software tool designed to calculate the effect of weight added to a weapon on the weapon's mass properties (COM, MOI, and total mass). The weapon COM tool was developed in support of the North Atlantic Treaty Organization (NATO) Research and Technology Organization (RTO) Systems Concepts and Integration (SCI)-178 RTO Task Group (RTG)-043 task group on dismounted Soldier system weapon systems inter-operability. The NATO SCI-178 RTG-043 task group has identified assault rifle weapon weight and COM as a primary research area.

Let f(X) and g(Y) be nondegenerate quadratic forms of dimensions m and n respectively over a field K, charK ≠ 2. Herein, the problem of the birational composition of f(X) and g(Y) is considered, namely, the condition is established when the product f(X)g(Y) is birationally equivalent over K to a quadratic form h(Z) over K of dimension m + n? The main result of this paper is the complete solution of the problem of the birational composition for quadratic forms f(X) and g(Y) over a field K when m = 2. The sufficient and necessary conditions for the existence of birational composition h(Z) for quadratic forms f(X) and g(Y) over a field K for m = 2 are obtained. The set of quadratic forms is described which can be considered as h(Z) in this case.

A full-rank lattice L in R is a discrete subgroup of R which is the set of all integer linear combinations of n-linearly independent vectors, say b1, · · · ,bn i.e., L = { ∑n i=1 zibi | z1, · · · , zn ∈ Z}. The matrix B = [b1, · · · ,bn] is called the basis of the lattice and the matrix Q = B′B is called a Gram matrix of the lattice. A lattice is integral if its Gram matrix has only integer entries. Integral lattices have been studied by mathematicians as positive definite quadratic forms, defined by the equation x′Qx, where x = (x1, · · · , xn) and Q ∈ Zn×n. One of the classical problems in this area is the classification of quadratic forms. Two quadratic forms are equivalent over Z if one can be obtained by the other using a unimodular transformation. If two quadratic forms are equivalent over Z then they are equivalent over the ring Z/pZ, for all primes p and positive integers k. The converse is not true. This leads to the classification of integral quadratic forms into equivalence classes, called the genus. A genus is a set of quadratic forms which are equivalent over Z/pZ for all primes p and positive integers k. The main result of this thesis is to generate a quadratic form of a given genus in randomized polynomial time. Of independent interest is a polynomial time algorithm to generate a uniform random solution of the equation x′Qx ≡ t mod p.

<p>Inspection of large space structures is imperative for long term mission success. One solution is to utilize a second free flying spacecraft capable of performing inspection in orbit. An Extended Kalman Filter (EKF) is used to perform estimations on the relative position, velocity, angular velocity, and attitude through the use of navigation markers. Simulation takes place using MATLAB to compare the true values with the estimated values using the EKF. The initial covariance (P), process noise covariance (Q), and measurement noise covariance (R) matrices were tuned for a space structure with three navigation points. The largest recorded errors over 100 iterations occurred during the initial estimation yielding 26.78 centimeters in relative position, 1.80 centimeters per second in relative velocity, 0.0444 radians per second in relative angular velocity, and a difference of 0.0172 in the unit vector of relative attitude. After allowing 20 seconds of settling time the maximum errors were reduced to 5.0 centimeters in relative position, 0.40 centimeters per second in relative velocity, 0.0046 radians per second in relative angular velocity, and a difference of 0.0020 in the unit vector of relative attitude. The paper also discusses the application of training algorithms to tune the EKF parameters for future consideration.</p>

In this study, a simple method based on the dynamic equation of motion was introduced to determine the moment of inertia using a commercial dynamometer, and an optimization technique was utilized to estimate inertial parameters with the determined moment of inertia. To evaluate the feasibility of the developed method, three different passive speeds (i.e. 240, 270 and 300°/s) were chosen to confirm whether the moment of inertia values are the same irrespective of angular speeds. Moreover, the estimated inertial parameters (i.e., the mass, center of mass and moment of inertia) of the elbow attachment and the disk-like 3 kg-weight were compared with solutions of uniform square cube and solid disk, respectively. As a result, the values of moments of inertia of the elbow attachment were 0.216 ± 0.017, 0.215 ± 0.016 and 0.216 ± 0.017 kg · m(2) at angular speeds of 240, 270 and 300°/s, respectively. The values of the moment of inertia of both the attachment and weight were 0.821 ± 0.054, 0.823 ± 0.058 and 0.824 ± 0.053 kg · m(2) at angular speeds of 240, 270 and 300°/s, respectively. There were no significant differences among the speeds. The estimated inertial parameters of the attachment or the weight were very similar to the theoretical values. Therefore, it is expected that the developed method has the potential to estimate inertial parameters of a human body segment and to improve the accuracy and reliability of the studies on human dynamics.

We present a versatile experimental apparatus for exploring rotational motion through the interplay between the moment of inertia, torque, and rotational kinetic energy of a wheel. The heart of this experiment uses a 3D-printed wheel along with easily accessible stock components that allow for the adjustment of the moment of inertia while keeping the total mass of the wheel constant. The wheel can act as a massive pulley of variable moment of inertia that allows students to measure the moment of inertia of the bare wheel by applying a constant torque to the system. The wheel can also be used to explore rotational kinetic energy in the form of races down ramps. The 3D-printed aspect of this wheel allows anyone with access to a 3D printer to create, explore, and modify this wheel at a low cost, allowing for more flexibility and accessibility for student and instructor exploration and modification. In the study of linear kinematics, it is easy to investigate how systems evolve with variable mass. With rotational motion the situation is more complicated due to the fact that the moment of inertia depends not only on the mass, but also on the distribution of the mass. This can be demonstrated in the form of massive pulleys in Atwood machines and in a rolling race of objects of varying mass and moment of inertia. For lab explorations it is best to isolate a single variable that can be changed. For rotational motion it is difficult to find a wheel with constant mass but different moments of inertia. There do exist examples that solve this problem, but they require workshop access and cannot act as pulleys. To address this issue we have designed a wheel whose moment of inertia can easily be created and manipulated while the mass of the system remains constant, providing an accessible, flexible, and robust apparatus to explore the interplay between moment of inertia and torque and rotational kinetic energy.

Soup Can Races: Teaching Rotational Dynamics Energy-based Solutions

Using space geodetic observations from four techniques (GPS, VLBI, SLR and DORIS), we simultaneously estimate the angular velocities of 11 major plates and the velocity of Earth's centre. We call this set of relative plate angular velocities GEODVEL (for GEODesy VELocity). Plate angular velocities depend on the estimate of the velocity of Earth's centre and on the assignment of sites to plates. Most geodetic estimates of the angular velocities of the plates are determined assuming that Earth's centre is fixed in an International Terrestrial Reference Frame (ITRF), and are therefore subject to errors in the estimate of the velocity of Earth's centre. In ITRF2005 and ITRF2000, Earth's centre is the centre of mass of Earth, oceans and atmosphere (CM); the velocity of CM is estimated by SLR observation of LAGEOS's orbit. Herein we define Earth's centre to be the centre of mass of solid Earth (CE); we determine the velocity of CE by assuming that the portions of plate interiors not near the late Pleistocene ice sheets move laterally as if they were part of a rigid spherical cap. The GEODVEL estimate of the velocity of CE is likely nearer the true velocity of CM than are the ITRF2005 and ITRF2000 estimates because (1) no phenomena can sustain a significant velocity between CM and CE, (2) the plates are indeed nearly rigid (aside from vertical motion) and (3) the velocity of CM differs between ITRF2005 and ITRF2000 by an unacceptably large speed of 1.8 mm yr−1. The velocity of Earth's centre in GEODVEL lies between that of ITRF2000 and that of ITRF2005, with the distance from ITRF2005 being about twice that from ITRF2000. Because the GEODVEL estimates of uncertainties in plate angular velocities account for uncertainty in the velocity of Earth's centre, they are more realistic than prior estimates of uncertainties. GEODVEL differs significantly from all prior global sets of relative plate angular velocities determined from space geodesy. For example, the 95 per cent confidence limits for the angular velocities of GEODVEL exclude those of REVEL (Sella et al.) for 34 of the 36 plate pairs that can be formed between any two of the nine plates with the best-constrained motion. The median angular velocity vector difference between GEODVEL and REVEL is 0.028° Myr−1, which is up to 3.1 mm yr−1 on Earth's surface. GEODVEL differs the least from the geodetic angular velocities that Altamimi et al. determine from ITRF2005. GEODVEL's 95 per cent confidence limits exclude 11 of 36 angular velocities of Altamimi et al., and the median difference is 0.015° Myr−1. GEODVEL differs significantly from nearly all relative plate angular velocities averaged over the past few million years, including those of NUVEL-1A. The difference of GEODVEL from updated 3.2 Myr angular velocities is statistically significant for all but two of 36 angular velocities with a median difference of 0.063° Myr−1. Across spreading centres, eight have slowed down while only two have sped up. We conclude that plate angular velocities over the past few decades differ significantly from the corresponding angular velocity averaged over the past 3.2 Myr.

Traditionally, the equations of motion of a solid body uses the theorem of the center of mass motion and the theorem of change of angular momentum with respect to the center of mass (Koenig’s axes). However, in the investigation of a solid body motion in a number of cases, the generalized coordinates are advisable to choose coordinates of another point of a body (pole) does not match the center of mass of the body, and the angles of rotation around this point. We obtain the equations of a solid body motion for these generalized coordinates. This is important for mathematical modeling of the motion of various objects, including vehicles. During the motion of a vehicle, its mass-inertial characteristics may change, including the position of the center of mass and the direction of the main axes of inertia, body mass and moments of inertia.

Isolating the Contributions from Moments of Inertia in Isotopic Shifts Measured by High-Resolution Cyclic Ion Mobility Separations.

The spatial momentum relation of an underac-tuated articulated multibody system on a floating base is a dynamic equilibrium relation between its coupling and relative momenta. The relative momentum is the difference between the system momentum and the momentum of the composite-rigid-body (CRB) that is obtained when the joints are locked. This relation is referred to as the momentum equilibrium principle. The focus in this work is on the angular momentum component of the momentum equilibrium principle. It is clarified that the relative angular momentum component can be represented in terms of the so-called relative angular velocity that is used as a control input in a balance controller. The balance controller proposed here is a whole-body controller that has independent inputs for center of mass (CoM) velocity and base-link angular velocity control. In addition, the relative angular velocity control input endows the controller with the unique property of generating an appropriate upper-limb motion that can stabilize the system momentum. More specifically, it is shown that when the relative angular velocity is derived from the reaction null-space (RNS) of the system, it becomes possible to stabilize the unstable states with a rolling foot/feet. The formulation is simple and yet quite efficient — there is no need to modify the contact model to account for the transitions between the stable and unstable contact states. There is also no need to command the upper-limb motion directly. A few simulation examples are presented to demonstrate and discuss the properties of the controller.

We consider the problem of controlling a spherical robot with a pendulum actuator rolling on a platform that is capable of moving translationally in the horizontal plane of absolute space. The spherical robot is subject to holonomic and nonholonomic constraints. Some point target moves at the level of the geometric center of the spherical robot and does not touch the moving platform itself. The motion program that allows the spherical robot to pursue a target is specified through two servo-constraints. The robot can follow a target from any position and with any initial conditions. Two ways to control this system in absolute space are proposed: by controlling the forced motion of the platform (the pendulum oscillates freely) and by controlling the torque of the pendulum (the platform is stationary or oscillates inconsistently with the spherical robot). The equations of motion of the system are constructed. In the case of free oscillations of the pendulum, the system of equations of motion has first integrals and, if necessary, can be reduced to a fixed level of these integrals. When a spherical robot moves in a straight line, for a system reduced to the level of integrals, phase curves, graphs of the distance from the geometric center of the spherical robot to the target, the trajectory of the selected platform point when controlling the platform, and the square of the control torque when controlling the pendulum drive are constructed. When the robot moves along a curved path, integration is carried out in the original variables. Graphs of the squares of the angular velocity of the pendulum and the spherical robot itself are constructed, as well as the trajectory of the robot’s motion in absolute space and on a moving platform. Numerical experiments were performed in the Maple software package.

Specification of mass properties is an essential step in the modeling of jaw dynamics, but obtaining them can be difficult. Here, we used three-dimensional computed tomography (CT) to estimate jaw mass, mean bone density, anatomical locations of the mass and geometric centers, and moments of inertia in the pig jaw. High-resolution CT scans were performed at one-mm slice intervals on specimens submerged in water. The mean estimated jaw mass was 12% greater than the mean wet weight, and 33% more than the mean dry weight. Putative bone marrow accounted for an extra 13% of mass. There was a positive correlation between estimated mean bone density and age. The mass center was consistently in the midline, near the last molar. The mean distance between the mass center and geometric center was small, especially when bone marrow was taken into account (0.58 +/- 0.21 mm), suggesting that mass distribution in the pig jaw is almost symmetrical with respect to its geometric center. The largest moment of inertia occurred around each mandible's supero-inferior axis, and the smallest around its antero-posterior axis. Bone marrow contributed an extra 9% to the moments of inertia in all three axes. Linear relationships were found between the actual mass and a mass descriptor (product of the bounding volume and mean bone density), and between the moments of inertia and moments of inertia descriptors (products of the mass descriptor and two orthogonal dimensions forming the bounding box). The study suggests that imaging modalities revealing three-dimensional jaw shape may be adequate for estimating the bone mass properties in pigs.

This study determines segmental dynamics parameters based on subject specific method. Five hemiplegic patients participated in the study, two men and three women. Their ages ranged from 50 to 60 years, weights from 60 to 70 kg and heights from 145 to 170 cm. Sample group included patients with different side of stroke. The parameters of the segmental dynamics resembling the knee joint functions measured via measurement of Winter and its model generated via the employment Kane’s equation of motion. Inertial parameters in the form of the anthropometry can be identified and measured by employing Standard Human Dimension on the subjects who are in hemiplegia condition. The inertial parameters are the location of centre of mass (COM) at the length of the limb segment, inertia moment around the COM and masses of shank and foot to generate accurate motion equations. This investigation has also managed to dig out a few advantages of employing the table of anthropometry in movement biomechanics of Winter’s and Kane’s equation of motion. A general procedure is presented to yield accurate measurement of estimation for the inertial parameters for the joint of the knee of certain subjects with stroke history.